Remark.
In the numerical examples of cyclic codes throughout
this paper, the \(n\) -root of unity \(\alpha \)  is fixed as follows.
Let \(\gamma \)  be the primitive element in \(\mathbb {F}_{q^r}\) 
chosen by Magma [1]},
then set \(\alpha =\gamma ^{(q^r-1)/n}\) .
